Efficient Nonparametric Estimation of Causal Effects in
Randomized Trials with Noncompliance
BY
Jing Cheng
Division of Biostatistics, University of Florida College of Medicine,
Gainesville, Florida 32610, U.S.A.
[email protected]
Dylan S. Small
Department of Statistics, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, U.S.A.
[email protected]
Zhiqiang Tan
Department of Statistics, Rutgers University,
Piscataway, New Jersey 08854, U.S.A
[email protected]
AND
Thomas R. Ten Have
Division of Biostatistics, University of Pennsylvania School of Medicine,
Philadelphia, Pennsylvania 19104, U.S.A.
SUMMARY
Causal approaches based on the potential outcome framework provide a useful tool for addressing noncompliance problems in randomized trials. We propose a new estimator of causal treatment effects in randomized clinical trials with noncompliance. We use the empir-ical likelihood approach to construct a profile random sieve likelihood and take into account the mixture structure in outcome distributions, so that our estimator is robust to paramet-ric distribution assumptions and provides substantial finite-sample efficiency gains over the standard instrumental variable estimator. Our estimator is asymptotically equivalent to the standard instrumental variable estimator, and it can be applied to outcome variables with a continuous, ordinal or binary scale. We apply our method to data from a randomized trial of an intervention to improve the treatment of depression among depressed elderly patients in primary care practices.
Some key words: Causal effect; Efficient nonparametric estimation; Empirical likelihood;
Noncompliance; Randomized trials.
1. Introduction
When there is noncompliance in randomized trials, there is often interest in estimat-ing the causal effect of actually receivestimat-ing the treatment compared to receivestimat-ing the control. Knowledge of this effect is useful for predicting the impact of the treatment in a setting for which compliance patterns might differ from the randomized trial and for scientific under-standing of the treatment (Sommer & Zeger, 1991; Sheiner & Rubin, 1995; Small et al., 2006; Cheng & Small, 2006).
Note that intention-to-treat analysis is not suitable for estimating the causal effect of actually receiving the treatment when there is noncompliance because it estimates the effect of assignment to the treatment group. An as-treated analysis seeks to estimate the causal effect of receiving the treatment but is biased if compliers are not comparable to noncom-pliers. Imbens & Angrist (1994) and Angrist et al. (1996) show that the causal effect of actually receiving the treatment for the subgroup of subjects who would receive the treat-ment if assigned to the treattreat-ment group and would receive the control if assigned to the
control group, called the complier average causal effect or the local average treatment ef-fect in the econometrics literature (Imbens & Angrist, 1994), is nonparametrically identified under certain, often plausible, assumptions that do not require compliers and noncompliers to be comparable. These assumptions, henceforth referred to as the instrumental variable assumptions, are discussed in §2. The complier average causal effect can be consistently esti-mated under the instrumental variable assumptions by the standard two-stage least-squares instrumental variables estimator. Imbens & Rubin (1997a, b) demonstrate that, under the assumptions, the standard instrumental variable estimator is an inefficient estimator of the complier average causal effect because it does not make full use of the mixture structure of the outcome distributions of the four observed groups defined by the cross classification of the randomization and treatment received; see §2.4 for further discussion. Imbens & Rubin (1997b) present three new alternatives to the standard IV estimator. One is based on a normal approximation and two are based on multinomial approximations to the outcome distributions in the four groups. In a simulation study with normally distributed outcomes, Imbens & Rubin (1997b) show that all three alternative estimators are more efficient than the standard IV estimator. However, the estimator that is based on a normal approximation to the outcome distributions can have substantial bias when the outcomes are not normal; this is demonstrated in §4. The estimators based on multinomial approximations to the outcome distributions are in principle nonparametric. However, a systematic approach for choosing the multinomial approximations is needed.
Multinomial approximations to the outcome distributions are a type of sieve. A sieve is a sequence of approximations {Fn} to a space F of distributions such that Fn→ F as n → ∞
(Grenander, 1981). Maximizing the likelihood over a sieve rather than the whole parameter space often leads to desirable statistical properties, especially when the underlying parameter space is large (Shen & Wong, 1994). However, the construction of sieves is not an easy task. One approach to constructing sieves is to use a random approximation ˆFn that depends on
the data, a random sieve. The empirical likelihood approach (Owen, 1991) is based on an easily constructed random sieve (Shen et al., 1999). In this paper, we use the empirical
likelihood approach to construct an efficient estimator for the complier average causal effect. 2. Notation, Assumptions and Review of Established Estimators
2.1 Notation
We consider a two-arm randomized trial with N subjects, n0 of whom are randomly
assigned to the control group. We use letters with and without star to denote vectors and scalars respectively. Let R∗ be the N-dimensional vector of randomization assignments
for all subjects, with individual element Ri = r ∈ {0, 1} according to whether subject i
is assigned active treatment, Ri = 1, or control, Ri = 0. We let Ar∗∗ be the N-dimensional
vector of potential treatment receiveds under the vector of randomization assignment r∗with
individual element Ar∗
i = a ∈ {0, 1} according to whether subject i would take the control
or treatment under randomization assignment r∗. We let Y∗r∗,a∗ be the vector of potential
responses under randomization assignment r∗ and treatment receiveds a∗, with individual
element Yr∗,a∗
i being the potential response for subject i with the vectors of randomization
assignments r∗ and treatment receiveds a∗. The sets of {Yir∗,a∗|r∗ ∈ {0, 1}N, a∗ ∈ {0, 1}N}
and {Ar∗
i |r∗ ∈ {0, 1}N} are ‘potential’ responses and treatment receiveds in the sense that
we can only observe one member of each set. The observed outcome and treatment received variables for subject i are YR∗,AR∗∗
i ≡ Yi and ARi ∗ ≡ Ai respectively.
2.2. Assumptions
We make similar assumptions to those in Angrist et al. (1996).
Assumption 1: Stable unit treatment value assumption (Rubin, 1980). (i). If r = r0, then
Ar∗
i = A
r0 ∗
i for subject i. (ii). If r = r0 and a = a0, then Yir∗,a∗ = Y r0
∗,a0∗
i for subject i. This
assumption allows us to write Yr∗,a∗
i , Ari∗ as Yir,a, Ari.
Assumption 2: Random assignment. This assumption implies independence between
assignment and pretreatment variables including potential outcomes and treatment receiveds.
Assumption 3: Random sampling. We assume that the N subjects in the trial are
independent and identically distributed draws from a superpopulation; that is, Yir,a and Ar i,
i = 1, . . . , N, are independent and identically distributed with the same distribution as the
Assumption 4: Mean exclusion restriction. We assume that E(Yr,a) = E(Yr0,a
) for all
r, r0, a; that is, the randomization assignment affects the mean of the observed outcome
only through its effect on treatment received. Note that the mean exclusion restriction is weaker than the unit level exclusion restriction of Angrist et al. (1996), who assume that
Yir,a= Yir0,a for all r, r0, a. However, we think that in most applications in which the weaker
mean exclusion restriction is plausible, the stronger unit-level exclusion restriction is also plausible and so we primarily use the weaker mean exclusion restriction because it is easier to work with this assumption.
Assumption 5: Nonzero average causal effect of R on A.
Assumption 6: Monotonicity. We assume that pr(A1 ≥ A0) = 1. This assumption says
that there is no one who would receive the opposite treatment of his or her assignment under both assignment to treatment and to control.
2.3 Compliance classes
A subject in a two-arm trial can be classified into one of four compliance classes:
Ci = 0 (never-taker) if (A0 i, A1i) = (0, 0) 1 (complier) if (A0 i, A1i) = (0, 1) 2 (always-taker) if (A0 i, A1i) = (1, 1) 3 (defier) if (A0 i, A1i) = (1, 0).
In practice, we can observe only one of A0
i and A1i, so that a subject’s compliance status is not
observed directly in a trial, but it can be partially identified based on treatment assignment and observed treatment-received; see Table 1. Note that the monotonicity assumption rules out the existence of defiers. For single consent design trials (Zelen, 1979), which have the property that the control group cannot access the treatment, that is, pr(A0 = 0) = 1, the
presence of always-takers and defiers is ruled out. 2.4 Major established estimators
Under Assumptions 1 − 6, the compliers are the only subgroup for which a randomized trial provides information about the causal effect of receiving treatment (Angrist et al., 1996).
For always-takers and never-takers, assignment to treatment has no effect on treatment received. The complier average causal effect, E(Y1− Y0|C = 1), can be thought of as the
causal effect of receiving treatment for the subpopulation of compliers because, for compliers, assignment of treatment agrees with receipt of treatment. Angrist et al. (1996) show that, under Assumptions 1-6, the complier average causal effect is
CACE= E(Y |R = 1) − E(Y |R = 0)
E(A|R = 1) − E(A|R = 0), (1)
which is the intention-to-treat effect divided by the proportion of compliers. The standard instrumental variable estimator is the sample analogue of (1),
ˆ CACES = ˆ E(Y |R = 1) − ˆE(Y |R = 0) ˆ E(A|R = 1) − ˆE(A|R = 0), (2)
where the ˆE’s denote sample means; (2) is sometimes called the Wald estimator.
The standard instrumental variable estimator does not take full advantage of the mixture structure of the outcomes of the four observed groups in Table 1, as we will discuss in §3.1. Imbens & Rubin (1997a,b) present two approaches of using mixture modeling to estimate the complier average causal effect. One approach assumes a parametric distribution, such as normal, for the outcomes for each compliance class group under each randomization assignment. The complier average causal effect is then estimated by maximum likelihood for this model using theEM algorithm. This estimator provides considerable efficiency gains over the standard instrumental variable estimator when the parametric assumptions hold; see Table 4. However, when the parametric assumptions are wrong, this estimator can be inconsistent whereas the standard instrumental variable estimator is consistent; see Table 4 for finite-sample results.
Imbens and Rubin’s other approach to using mixture modeling to estimate the complier average causal effect is to approximate the density of the outcome distribution for each compliance class under each randomization group as a piecewise constant function, and then estimate the complier average causal effect by maximum likelihood. This approach is
in principle nonparametric as the number of constant pieces in each density function can be increased with the sample size. However, Imbens & Rubin (1997b) do not provide a systematic approach for choosing the number of and locations of the pieces. We develop a systematic easily implementable approach for doing this using empirical likelihood in the next section.
3. Estimation through Empirical Likelihood Approach 3.1 Motivation and description of empirical likelihood approach
We first motivate and describe our method for single consent design trials, where the presence of always-takers and defiers is ruled out. Table 2 shows the relationship between observed (R, A) groups and latent compliance classes for a single consent design trial. The complier average causal effect can be re-expressed under Assumptions 1-6 as follows:
CACE= µc1− µc0 = µc1−µR=0− (1 − πc)µn
πc
=
E(Y |R = 1, A = 1) −E(Y |R = 0) − {1 − pr(A = 1|R = 1)}E(Y |R = 1, A = 0)
pr(A = 1|R = 1) (3)
where µc1, µc0, µnand µR=0denote the mean potential outcomes of the compliers under
treat-ment, compliers under control, never takers and the whole population of subjects when as-signed to the control respectively; and πcdenotes the proportion of compliers. The standard
instrumental variable estimator estimates the complier average causal effect by substituting the method of moments estimates from the sample for E(Y |R = 1, A = 1), E(Y |R = 0), pr(A = 1|R = 1) and E(Y |R = 1, A = 0) into (3). However, as noted by Imbens & Rubin (1997b), there are restrictions on the joint density of (Y, R, A) that are not taken into account by the method of moments that can be useful for estimating E(Y |R = 0), pr(A = 1|R = 1) and E(Y |R = 1, A = 0). To be specific, Assumptions 1-6 imply the following restrictions.
Restriction 1. The distribution of Y |R = 0 is a mixture of the outcome distribution of
the never-takers under R = 0 and the outcome distribution of the compliers under R = 0.
Restriction 2. The mixing proportion πc for Y |R = 0 equals pr(A = 1|R = 1) as a
consequence of Assumption 2.
Restriction 3. The mean of the never-takers under R = 0 is equal to the mean of the
never-takers under R = 1, which equals E(Y |R = 1, A = 0), as a consequence of Assumption 4.
The sample mean of Y |R = 0 uses only the information in those of Y1, . . . , YN for which
Ri = 0 to estimate E(Y |R = 0), but Restrictions 1-3 imply that there is additional
in-formation in those of Y1, . . . , YN for which Ri = 1. Similarly, the sample proportion of
A = 1|R = 1 uses only the information in those of A1, . . . , AN for which Ri = 1 to estimate
pr(A = 1|R = 1) but Restrictions 1-3 imply that there is additional information in those of A1, . . . , AN for which Ri = 0. A body of work has shown that supplementing a sample
from a distribution that is a mixture of two components with samples from one or both of the components alone provides additional information for estimating aspects of the mixture distribution; see for example Hall & Titterington (1984), Lancaster & Imbens (1996) and Qin (1999). Here, the sample of Y1, . . . , YN for which Ri = 1, Ai = 0 provides information
about the never-taker component of the mixture Y |R = 0 and the sample of A1, . . . , AN for
which Ri = 1 provides information about the mixing proportion in the mixture Y |R = 0.
We now illustrate how this information is useful in a setting with a binary outcome in which
πc= 0.5, µn = 0.2, µc1 = 0.8, µc0 = 0.9, N = 40, n0 = 20. (4)
The following is a plausible sample in this setting: #(Yi = 1, Ai = 1, Ri = 1) = 8, #(Yi =
0, Ai = 1, Ri = 1) = 2, #(Yi = 1, Ai = 0, Ri = 1) = 2, #(Yi = 0, Ai = 0, Ri = 1) = 8,
#(Yi = 1, Ai = 0, Ri = 0) = 13 and #(Yi = 0, Ai = 0, Ri = 0) = 7; the p-value for a χ2 test of
whether or not this sample comes from the distribution (4) is 0.37. Note that, for this sample, the method of moments estimates of the quantities in (3), namely ˆE(Y |R = 1, A = 1) = 0.8,
ˆ
E(Y |R = 0) = 0.65, ˆpr(A = 1|R = 1) = 0.5, ˆE(Y |R = 1, A = 0) = 0.2, violate Restrictions
1-3. Figure 1 plots the profile log-likelihood for this sample under the probability model given by Assumptions 1-6 with binary outcomes. The maximum likelihood estimator of
1-3, has a noticeably higher likelihood than the standard instrumental variable estimator, which ignores some of the restrictions. The maximum likelihood estimator’s property of taking into full account the mixture structure leads to substantially better estimates; in 1000 simulations from model (4), the mean squared error of the maximum likelihood estimator was 0.048 compared to 0.156 for standard instrumental variable estimator.
To take account of the mixture structure of the outcomes given by Restrictions 1-3 for more general distributions of outcomes in a nonparametric way, we use the empirical likelihood approach. The empirical likelihood for a parameter such as the complier average causal effect is the nonparametric profile likelihood for the parameter. Maximum empirical likelihood estimators have good properties for a wide class of semiparametric problems; see Owen (2001) and Qin & Lawless (1994) for discussion.
Without loss of generality, we arrange the subjects so that R1 = . . . = Rn0 = 0 and
Rn0+1 = . . . = RN = 1; thus, (Y1, A1), . . . , (Yn0, An0) is a random sample from the population
of Yr=0,a=Ar=0
, Ar=0and (Y
n0+1, An0+1), . . . , (YN, AN) is a random sample from the population
of Yr=1,a=Ar=1
, Ar=1. The empirical likelihood L
E of the parameters (πc, µn, µc1, µc0) is:
LE(πc, µn, µc1, µc0) = max à n0 Y i=1 qi ! à N Y i=n0+1 qi ! , (5) subject to n0 X i=1 qi = 1, N X i=n0+1 qi = 1, qi ≥ 0, i = 1, . . . N, (6) N X i=n0+1 qiAi = πc, N X i=n0+1 qiYiAi = µc1πc, N X i=n0+1 qiYi(1 − Ai) = µn(1 − πc), (7) There exist pc0 i , pni, i = 1, . . . , n0 such that πcpc0i + (1 − πc)pni = qi, (8) n0 X i=1 pc0 i = n0 X i=1 pn i = 1, pc0i , pni ≥ 0, i = 1, . . . , n0, (9) n0 X i=1 pn i(Yi− µn) = 0, (10) n0 X i=1 pc0 i (Yi− µc0) = 0. (11) 8
Note that throughout our paper, we will follow Owen (2001, Ch. 2.3) and regard tied data values Yi, Yj as representing distinct outcomes in the empirical likelihood as this
sim-plifies calculations and does not affect inferences. The pc0
i and pni in (8)-(11) represent
the population probabilities that a complier assigned to the control and a never-taker as-signed to the control have the same outcome as subject i respectively. The conditions (8)-(11) involving the pc0
i and pni encode the restrictions on the distribution of Y |R = 0
that come from it being a mixture of the compliers and never-takers under Assumptions 1-6, see Restrictions 1-3. The maximum empirical likelihood estimate of (πc, µn, µc1, µc0)
is arg maxπc,µn,µc1,µc0LE(πc, µ
n, µc1, µc0). To ease the computational burden of computing
the maximum empirical likelihood estimate, we do not maximize over µn, but instead use
the method of moments estimator ˆµn =PN
i=1YiRi(1 − Ai)/
PN
i=1Ri(1 − Ai) and maximize
LE(πc, ˆµn, µc1, µc0) over (πc, µc1, µc0). In model (4), this approximate maximum empirical
likelihood estimator of the complier average causal effect performed almost as well as the maximum empirical likelihood estimator; its mean squared error was 0.051 compared to 0.048 for the maximum empirical likelihood estimator. We now present an algorithm for finding the approximate maximum empirical likelihood estimate.
3.2 Computation for empirical likelihood approach
To find the approximate maximum empirical likelihood estimate, we conduct a grid search over πc, finding maxµc1,µc0LE(˜πc, ˆµn, µc1, µc0) over a grid of ˜πcfrom 0 to 1. As we will see be-low, arg maxµc1LE(˜πc, ˆµn, µc1, µc0) does not depend on µc0and arg maxµc0LE(˜πc, ˆµn, µc1, µc0) does not depend on µc1, so finding the maximizing µc1 and µc0 can be done separately. For
finding the maximizing µc1, we note that arg max
µc1LE(˜πc, ˆµn, µc1, µc0) equals arg maxµc1 Q
i:Ri=Ai=1qi
subject to (i) Pi:Ri=Ai=1qi = ˜πc, (ii) qi ≥ 0, i = 1, . . . , N and (iii)
P
i:Ri=Ai=1qiYi = ˜πcµ
c1.
By multiplying the qi’s by 1/˜πc, we see that finding arg maxµc1LE(˜πc, ˆµn, µc1, µc0) is equiv-alent to finding the maximum empirical likelihood estimator of the mean of the popula-tion of Y1,1|C = 1 based on the random sample Y
1, . . . , Yn|Ai = 1, Ri = 1; consequently,
arg maxµc1LE(˜πc, ˆµn, µc1, µc0) is the mean of Y1, . . . , Yn|Ai = 1, Ri = 1; see Theorem 2.1 of Owen (2001). Thus, our estimate of µc1 is ˆµc1 = ³PN
i=1RiAi
´−1P
N
our estimate of µc0, let (q∗
1, . . . , qn∗0) = arg maxq1,...,qn0
Qn0
i=1qi subject to (6) and (8)-(10) with
µn = ˆµn, π c= ˜πc. We have that arg max µc0 LE(˜πc, ˆµ n, µc1, µc0) = Pn0 i=1q∗iYi − (1 − ˜πc)ˆµn ˜ πc , (12)
where we use the fact that, for the µc0 that satisfiesPn0
i=1qi∗Yi = ˜πcµc0+ (1 − ˜πc)ˆµn, the
con-straints (8)-(11) are satisfied for q1 = q1∗, . . . , qn0 = qn∗0. Thus, to find arg maxµc0LE(˜πc, ˆµn, µc1, µc0), we just need to find q∗
1, . . . , q∗n0. To do this, we note that we can view (q
∗
1, . . . , qn∗0) as the
maximum likelihood estimate of the category probabilities for the sample Y1, . . . , Yn0 from
an independent and identically distributed multinomial model with categories Y1, . . . , Yn0,
corresponding category probabilities q1, . . . , qn0 and parameter restrictions given by (6) and
(8)-(10) with µn = ˆµn, π
c = ˜πc. Finding the maximum likelihood estimate directly is
chal-lenging because of the complex parameter restrictions in (8)-(10). However, consider using the EM algorithm, where we regard each subject’s compliance class as ‘missing data.’ We can reexpress the observed data likelihoodQn0
i=1qiand the parameter restrictions (6) and (8)-(10)
in terms of pc0
i and pni; see Appendix 1 for details. We can then use the EM algorithm to find
the pc0
i and pni to maximize the observed data likelihood and then find the corresponding
max-imizing qi’s by (8). The complete-data likelihood is
Q i:Ri=0,Ci=1p c0 i Q i:Ri=0,Ci=0p n i. Since the
complete data follows an exponential family distribution, the E-step has a closed form expres-sion. The M-step involves a calculation analogous to finding the empirical likelihood for the mean (Owen, 1988); convex duality enables us to avoid maximizing over pc0
i , pni, i = 1, . . . , n0,
and instead we maximize over a single variable. The tractability of both the E- and M- steps makes the EM algorithm with each subject’s compliance class as missing data easy to use for finding q∗
1, . . . , qn∗0 and hence finding arg maxµc0LE(˜πc, ˆµ
n, µc1, µc0) by (12).
Note that, given qi, i = 1, . . . , n0, there are typically more than one set of pc0i , pni, i =
1, . . . , n0, that satisfy the constraints (8)-(10). Numerical experiments, not shown here,
verify that although the EM algorithm converges to different values of pc0
i and pni for different
sets of starting values for the pc0
i and pni, the corresponding qi’s to which the EM algorithm
converges are the same, as Lemma 1 shows more formally. Lemma 1. Regardless of the starting values for the pc0
i , pni, i = 1, . . . , n0, the sequence of
estimates of qi from the EM algorithm converges to the global maximum of the likelihood
Qn0
i=1qi subject to the restrictions (6) and (8)-(10) with µn= ˆµn, πc= ˜πc.
The proof of Lemma 1 is outlined in Appendix 2.
In summary, we estimate πc, µn, µc1, µc0 as follows; a program is available from the authors.
Step 1. We obtain ˆµn as the sample mean of Y |R = 1, A = 0.
Step 2. We obtain ˆµc1 as the sample mean of Y |R = 1, A = 1.
Step 3. For a grid of ˜πc, we find the maximum empirical likelihood estimate of µc0
given πc = ˜πc, µn = ˆµn, µc1 = ˆµc1 using the EM algorithm described above. Then ˆπc =
arg max˜πcmaxµc0LE(˜πc, ˆµn, ˆµc1, µc0) and ˆµc0 = arg maxµc0LE(ˆπc, ˆµn, ˆµc1, µc0).
Step 4. Our approximate maximum empirical likelihood estimate of the complier average
causal effect is CACEˆ A= ˆµc1− ˆµc0.
3.3 Estimation in trials in which the assigned to control group can access the treatment
Our method illustrated in §3.1 can be directly applied to more general trials under As-sumptions 1-6 in which the control group can access the treatment. For such trials, we have one more compliance class, the always-takers, in addition to the compliers and never-takers, see Table 3; we denote the proportion of always takers and the mean of always takers’ po-tential outcomes by πa and µa respectively. The empirical likelihood LE of the parameters
(πc, πa, µn, µa, µc1, µc0) is the maximum likelihood for multinomial distributions (q1, . . . , qn0)
on (Y1, A1), . . . (Yn0, An0) and (qn0+1, . . . , qN) on (Yn0+1, An0+1), . . . , (YN, AN) that are
con-sistent with (πc, πa, µn, µa, µc1, µc0) and the restrictions on the parameter space specified by
Assumptions 1-6, namely LE(πc, πa, µn, µa, µc1, µc0) = max (
Qn0 i=1qi) ³QN i=n0+1qi ´ subject to (i) Pn0 i=1qi = 1, PN
i=n0+1qi = 1; (ii) qi ≥ 0, i = 1, . . . , N; (iii)
P
i:Ri=1,Ai=0qi = 1 − πa− πc; (iv)Pi:Ri=0,Ai=1qi = πa; (v)
P i:Ri=1,Ai=0qiYi = µ n(1−π a−πc); (vi) P i:Ri=0,Ai=1qiYi = µ aπ a;
(vii) There exist pc0
πa−πc)/(1−πa)}pni = qi, (viib) P pc0 i = P pn i = 1, (viic) pc0i , pni ≥ 0, (viid) P pn i(yi−µn) = 0 and (viie) Ppc0
i (Yi − µc0) = 0; and (viii) There exist pc1i , pai for the i with Ri = 1, Ai = 1
such that (viiia) {πc/(πc + πa)}pc1i + {πa/(πc + πa)}pai = qi, (viiib)
P pc1 i = P pa i = 1; (viiic) pc1 i , pai ≥ 0; (viiid) P pa i(Yi − µa) = 0 and (viiie) P pc1 i (Yi − µc1) = 0. As with
the single consent design, rather than finding the maximum empirical likelihood estimate of (πc, πa, µn, µa, µc1, µc0), we find the approximate maximum empirical likelihood estimate
by setting µn equal to the sample mean of Y |R = 1, A = 0, corresponding to the known
never-takers in the sample, and µaequal to the sample mean of Y |R = 0, A = 1,
correspond-ing to the known always-takers in the sample, and then maximizcorrespond-ing the empirical likelihood over (πc, πa, µc1, µc0). This can be done by using the EM algorithm for estimating µc0 in the
Y |R = 0, A = 0 sample as in §3.2, and an analogous EM algorithm for estimating µc1 in the
Y |R = 1, A = 1 sample. The details are provided in a technical report available from the
authors.
4 Simulation Studies
We compare our approximate maximum empirical likelihood estimator with the standard instrumental variable estimator and Imbens and Rubin’s parametric estimator, considering single consent design trials as discussed in §3.1. We set πc = 0.5 and compare the three
estimators under different outcome distributions and under sample sizes of N = 100 and
N = 500 with pr(R = 1) = 0.5. The outcome distributions we consider are Normal, gamma,
and lognormal distributions. For each outcome distribution, we set µc1= 2, µc0= 1, so that
the CACE= µc1− µc0 = 1. The variances are fixed at 1.
Before explaining our settings for µn, we discuss the impact of the distance between µn
and µc0on the efficiency of the approximate maximum empirical likelihood estimator relative
to standard instrumental variable estimator. The distance between µnand µc0is a measure of
the separation between the distributions of the compliers and never-takers under the control. To see the impact of the distance between µnand µc0, we consider under what conditions the
approximate maximum empirical likelihood and standard instrumental variable estimators are equal. Standard instrumental variable estimator estimates the complier average causal
effect by substituting method of moments estimates into (3). The approximate maximum empirical likelihood estimator estimates the complier average causal effect by substituting maximum empirical likelihood estimates into (3) conditional on E(Y |R = 1, A = 0) being set equal to its method of moments estimate. The approximate maximum empirical likelihood estimator equals the standard instrumental variable estimator if the method of moments estimates of pr(A = 1|R = 1) and E(Y |R = 1, A = 0), denoted by ˆpr(A = 1|R = 1) and
ˆ
µn respectively, satisfy (8)-(10) with q
i = 1/n0 for i = 1, . . . , n0. This will happen if and
only if ˆµn is between the trimmed mean of Y |R = 0 over the 0 to {1 − ˆpr(A = 1|R = 1)}
quantiles and the trimmed mean of Y |R = 0 over the ˆpr(A = 1|R = 1) to 1 quantiles. It is more likely that ˆµn will escape these bounds when the distributions of the compliers and
the never-takers are more separated. When ˆµn does escape these bounds, we expect that
the approximate maximum empirical likelihood estimator will provide a better estimate than standard instrumental variable estimator because the approximate maximum empirical likelihood estimator is taking better account of the mixture structure of outcomes implied by Assumptions 1-6. Thus, we expect that the approximate maximum empirical likelihood estimator will gain more efficiency over standard instrumental variable estimator when the distance between µn and µc0 is greater, because then the distributions of the compliers and
never-takers under the control are more separated.
To see the effect of the separation between the compliers and never-takers under the control, we chose two sets of values for µc0and µnsuch that the distributions of the compliers
and never-takers under the control are well separated under one set of values but are close to each other under another set of values. In setting N1, the distributions of Y1,1
i |Ci =
1, Yi0,0|Ci = 1 and Yi0,0 = Yi1,0|Ci = 0 are Normal with (mean, variance) combinations
(2, 1), (1, 1) and (3, 1), respectively. In setting G1 and LN1, the distributions are gamma
and lognormal, respectively. Settings N2, G2and LN2differ only in that the (mean, variance)
combination of Yi0,0 = Yi1,0|Ci = 0 is (1.5, 1).
For each setting, we present summary results over 1000 replications with sample sizes of 100 and 500. Table 4 shows the bias and mean squared error from the three different
estimators for the complier average causal effect for the different settings considered. Table 4 shows the following features.
First, the parametric estimator based on the normality assumption is unbiased and more efficient than standard instrumental variable and approximate maximum empirical likelihood estimators under the true normal distributions, but shows biases of 23% − 40% and is less efficient than other two estimators under nonnormal distributions.
Secondly, both the approximate maximum empirical likelihood and standard instrumental variable estimators have low bias for all settings considered. The approximate maximum empirical likelihood estimator has bias below 5% when the distributions of the never-takers and the compliers under the control are close to each other. When the distributions of the never-takers and compliers under the control are well separated and the sample size is 100, the approximate maximum empirical likelihood estimator has a bias of about 10% but this bias drops to below 5% when the sample size increases to 500.
Thirdly, the approximate maximum empirical likelihood estimator is more efficient than standard instrumental variable estimator for all settings considered. The gain in mean squared error is more substantial when the distributions of never-takers and compliers under the control are well separated, as expected from the discussion above. The gain in mean squared error is as large as 56%. The gain is generally smaller with a sample size of 500 rather than 100. In additional simulations not presented in Table 4, we found that there is still a gain in mean squared error with the approximate maximum empirical likelihood estimator over standard instrumental variable estimator with a sample size of 1000.
We also did a simulation study for the setting of §3.3 in which the assigned to control group can access the treatment. The results are not presented, but are available from the authors. The pattern of results is similar to that for the single consent design trials.
5 Asymptotic Properties
In §4, we showed that the approximate maximum empirical likelihood estimator gains over standard instrumental variable estimator in a range of finite-sample situations, with larger gains when the compliers and never-takers’ outcome distributions under the control
are more separated. The standard instrumental variable estimator is based on estimating the distribution of (Y, A, R) by the empirical distribution of (Y, A, R); the method of moments estimators on which standard instrumental variable estimator is based are the moments of the empirical distribution. The source of the approximate maximum empirical likeli-hood estimator’s gain over standard instrumental variable estimator is that the empirical distribution of (Y, A, R) might not satisfy the restrictions given by Assumptions 1-6. The approximate maximum empirical likelihood estimator takes these restrictions into account to provide a better estimate of the distribution of (Y, A, R) than the empirical distribution. However, unless the distribution of (Y, A, R) is ‘at the boundary’ of the restrictions given by Assumptions 1-6, the empirical distribution of (Y, A, R) should satisfy the restrictions with probability converging to 1 as the sample size N → ∞. Consequently, the approximate maximum empirical likelihood estimator will be asymptotically equivalent to the standard instrumental variable estimator. We establish this result in Theorem 1 under condition (13) below. Condition (13) specifies that the distribution of (Y, A, R) is not ‘at the boundary’ of the restriction that the Y |R = 0 is a mixture of the compliers and never-takers under the control in the sense that the distributions of the compliers and never-takers under the control overlap at least minimally. In condition (13) below, we let Fc0 and Fn0 denote
the cumulative distribution functions of potential outcomes under the control for compliers and never-takers respectively, and we let G = πcFc0 + (1 − πc)Fn0 denote the cumulative
distribution function of potential outcomes under the control. The condition is 1 1 − πc Z G−1(1−π c) −∞ zdG(z) <R−∞∞ zdFn0(z) = µn, µn= Z ∞ −∞ zdFn0(z) < 1 1−πc R∞ G−1(πc)zdG(z). (13) Condition (13) says that the trimmed mean of the πn-smallest part of the mixture of
never-takers and compliers is strictly less than the mean of the never-never-takers and that the trimmed mean of the πn-largest part of the mixture of never-takers and compliers is strictly greater
Theorem 1. Consider a single consent design. Suppose (i) (13) holds, (ii) 0 < πc< 1 and
(iii) n0/N = d, 0 < d < 1. Then, pr(CACEˆ A=CACEˆ S) → 1 as N → ∞.
The proof of Theorem 1 is in Appendix 2.
In spite of the asymptotic equivalence result in Theorem 1, the simulation study in §4 showed that the approximate maximum empirical likelihood estimator can provide substan-tial gains in practical situations. The gains provided by the approximate maximum empirical likelihood estimator are analogous to the gains provided in estimating a population mean in the knowledge of restrictions on the range of the mean. For example, consider estimating the mean µ of a normal distribution N(µ, σ2) based on a random sample Y
1, . . . , YN when
it is known that µ is less than or equal to an upper bound µU. If µ is reasonably close to
µU, then the maximum likelihood estimate will gain substantially over the sample mean, the
maximum likelihood estimate if µ is unrestricted, for many sample sizes. However, as long as µ is less than µU by any amount, the estimators are equivalent asymptotically because,
for large enough N, the sample mean is less than µU with high probability.
6 Application to Depression Study
In this section, we apply our method to analyze a randomized trial of an intervention to improve treatment of depression among depressed elderly patients in primary care practices (Bruce et al., 2004). The encouragement intervention was that a depression care specialist collaborated with the patient’s primary care physician to facilitate adherence to a depression treatment strategy and provide education and assessment to the patient. The control was usual care. The study involved 539 depressed patients in 20 primary care practices at three sites followed for six visits: baseline, 4, 8, 12, 18 and 24 months. Each practice was randomized to either intervention, treatment, or usual care, control. For illustrative purposes, we ignore the fact that the trial was a group randomized trial and treat it as a completely randomized trial; for analyses that account for the group randomization, see Small et al. (2007). Compliance with the intervention was categorized as a binary variable, whether or not a patient had seen a depression care specialist in the prior four months of
follow-up. Patients in practices randomized to the usual-care group did not have access to the depression specialist, so there are only compliers and never-takers in this trial. To see the effects of estimators under different situations, we analyze two outcomes. One is the patients’ Hamilton depression scores measured at 4 months, which take integer values between 0 and 50. A lower value of the outcome means less depression. Another outcome of analysis is the composite anti-depression scores among males at one site measured at 12 months. This is an integer-valued score from 0 to 4 that indicates how much the patient is being treated for depression. A score of 3 or 4 is considered adequate treatment for depression while 1 or 2 means the patient is being treated in some way, but not a what is considered an adequate dose.
Table 5 shows the three estimates of the complier average causal effect for the Hamilton and composite anti-depression scores described above. The percentile bootstrap with 1000 resamples was used to compute approximate 95% confidence intervals. We first consider the Hamilton score at 4 months; see the second column of Table 5. The scores were observed for 517 subjects and 92.7% of these subjects that were assigned to treatment complied with the treatment. All the complier average causal effect estimates are negative and the 95% confidence intervals do not include zero, indicating that the intervention has a significant ben-eficial effect on depression compared to usual care. Comparing the three estimation methods, we first note from the histograms of the Hamilton outcome in Fig. 2 (a)-(c) that the Hamil-ton scores for the never-takers and compliers under the treatment are far from normally distributed, suggesting that the parametric estimator based on the normality assumption is probably a biased estimator. The standard instrumental variable estimator and the ap-proximate maximum empirical likelihood estimator provide very similar point estimates and similar 95% confidence intervals; see below for more explanation of this similarity. We now consider the outcome of the composite anti-depression scores among males at the site at 12 months, given in the third column of Table 5. The scores were observed for 37 subjects and 75% of these subjects who were assigned to treatment complied with the treatment. The approximate maximum empirical likelihood and standard instrumental variable
com-plier average causal effect estimates show a significant beneficial effect of the intervention on treating depression while the parametric normal estimate does not show a significant effect. As for the Hamilton score, the histograms of the composite anti-depression outcomes in Fig. 2 (d)-(f) show that the composite anti-depression scores from the never-takers and compliers under the control are far from normally distributed, suggesting that the parametric estima-tor based on the normality assumption is a biased estimaestima-tor. Unlike for the Hamilton score, for the complier average causal effect of the intervention on the composite anti-depression score, the approximate maximum empirical likelihood estimate has a substantially narrower 95% confidence interval than standard instrumental variable estimate.
The greater gain in efficiency of the approximate maximum empirical likelihood estimate compared to standard instrumental variable estimate for the composite anti-depression study rather than the Hamilton study is related to three factors. First, the sample size in the R = 0 group is smaller for the composite anti-depression study, making it more likely that the em-pirical distribution of (Y, A, R) will deviate from the restrictions implied by Assumptions 1-6. Secondly, the compliance rate among the subjects assigned to treatment is higher for the Hamilton study, 93%, than the composite anti-depression study, 75%, providing less scope in the Hamilton study for the extra information about Assumptions 1-6 used by the approximate maximum empirical likelihood estimator to have an impact. Thirdly, the sep-aration between the never-takers’ and compliers’ outcome distributions in the control group is greater for the composite anti-depression than for the Hamilton; if we use the estimates of
µn and µc0 obtained by substituting method of moments estimates into the population
ex-pressions for these quantities in (3), the estimated absolute standardized difference between the never-takers’ and compliers’ means in the control group is 2.34 for the composite anti-depression compared to 0.72 for the Hamilton. As we have shown in our simulation studies, the approximate maximum empirical likelihood estimator will have a larger gain in efficiency over standard instrumental variable estimator when the distributions of the never-takers and compliers in the control group are more separated.
7 Discussion 18
Our method can be extended to observational studies in which a variable R which en-courages, R = 1, or does not encourage, R = 0, a subject to take the treatment is not randomly assigned but is ‘as good as randomly assigned’, that is, ignorable, conditional on some covariates; such studies are discussed in Abadie (2003) and examples are given in Table 1 of Angrist & Krueger (2001). Suppose we replace Assumption 2 with Assumption 20 that the encouragment variable R is independent of Y1,1, Y1,0, Y0,1, Y0,0, A0, A1
condi-tional on a subject’s covariate vector X and that the encouragement variables of different subjects are independent. Also, suppose we expand Assumption 3 to Assumption 30 that
Xi, Yi1,1, Y 1,0 i , Y 0,1 i , Y 0,0
i , A0i, A1i are independent and identically distributed draws from a
su-perpopulation and expand Assumption 4 to condition on covariates, i.e, let Assumption 40
be that E(Yr,a|X) = E(Yr0,a
|X) for all r, r0, a, X. Furthermore, for a single consent
de-sign, suppose we consider linear models for the expected potential outcomes in a compliance class given the covariates and a logistic model for compliance given the covariates, i.e.,
E(Y1,1|C = 1, X) = X0βc1, E(Y0,0|C = 1, X) = X0βc0, E(Y1,0|C = 0, X) = E(Y0,0|C =
0, X) = X0βn and pr(C = 1|X) = expit(X0α), where expit(z) = ez/(1 + ez). We
in-clude an intercept in the covariate vector X and let p denote the dimension of X. Un-der this model, the complier average causal effect for compliers with covariate vector X is X0βc1− X0βc0. Under Assumptions 1, 20, 30, 40, 5 and 6 and the above models for the
outcomes and compliance probabilities, we have that the empirical likelihood of α, βc1, βc0
and βn is L
E(α, βn, βc1, βc0) = maxq1,...,qN QN
i=1qi subject to (i)
Pn0 i=1qi = 1, PN i=n0+1qi = 1; (ii) qi ≥ 0, i = 1, . . . , N; (iii) PN i=n0+1qiXij{Ai − expit(X 0 iα)} = 0, j = 1, . . . , p; (iv) PN i=n0+1qiAiXij(Yi − X 0 iβc1) = 0, j = 1, . . . , p; (v) PN i=n0+1qi(1 − Ai)Xij(Yi − Xi0βn) = 0,
j = 1, . . . , p; (vi) there exist tc0
i , tni, i = 1, . . . , n0such that (via) tc0i +tni = qi; (vib) tc0i , tni ≥ 0;
(vic)Pn0 i=1tc0i + Pn0 i=1tni = 1; (vid) Pn0 i=1tc0i Xij{1−expit(Xi0α)}+ Pn0 i=1tniXij{−expit(Xi0α)} = 0; (vie) Pn0
i=1tniXij(Yi − Xi0βn) = 0, j = 1, . . . , p; and (vif)
Pn0
i=1tc0i Xij(Yi − Xi0βc0) = 0,
j = 1, . . . , p. Here the tc0
i , tni, respectively represent the population probabilities that a
subject assigned to the control has the same outcome and covariates as subject i and is a complier, never-taker respectively. The above expression for the empirical likelihood builds
on Owen’s (2001, Ch. 4) discussion of empirical likelihood for regression models. As in our method of §3, we can compute the approximate maximum empirical likelihood estimate by estimating βn using the R = 1, A = 0 sample and maximizing the empirical likelihood over
α, βc1 and βc0 given βn = ˆβn.
When deriving the approximate maximum empirical likelihood estimator, we have as-sumed the weak exclusion restriction that the never-takers’, always takers’, respectively, mean is the same under assignment to treatment and control, rather than the strong exclu-sion restriction that the never-takers’, always- takers’, respectively entire outcome distribu-tion is the same under assignment to treatment and control. In most situadistribu-tions in which the weak exclusion restriction is plausible, we think that the strong exclusion restriction will also be plausible. We are currently adapting our approach to situations in which the strong exclusion restriction is plausible by enabling the empirical likelihood approach to use more equality constraints for aspects of the never-takers and always-takers under R = 0 and R = 1 distributions respectively than just equality of means.
ACKNOWLEDGEMENT
We thank the associate editor, a referee and Professor D.M. Titterington for their helpful comments and suggestions.
APPENDIX 1
Details of the EM algorithm
Reexpressing the observed data likelihoodQn0
i=1qi and the parameter restrictions (6) and
(8)-(10) in terms of pc0
i , pni, we have that the observed data likelihood, with πc= ˜πc, µn= ˆµn,
is Qn0
i=1{˜πcpc0i + (1 − ˜πc)pni}, with parameter restrictions n0 X i=1 pc0i = n0 X i=1 pni = 1, pc0i ≥ 0, pni ≥ 0, i = 1, . . . , n0, n0 X i=1 pni(Yi− ˆµn) = 0. (A1) where qi = ˜πcpc0i + (1 − ˜πc)pni and µc0 = Pn0
i=1pc0i Yi. Note that, if ˆµn is such that there is
no pc0
i , pni that satisfies (A1), then our approximate maximum empirical likelihood estimator
does not exist; in this case we can modify the approximate maximum empirical likelihood 20
estimator to use ˆµn as the closest point to P
i:Ri=1,Ai=0Yi/#{Ri = 1, Ai = 0}, the usual estimate of µnfor the approximate maximum empirical likelihood estimator, such that there
exists pc0
i , pni, i = 1, . . . , n0, that satisfy (A1). If we view each subject’s compliance class as
missing data, the complete data likelihood is Qi:Ri=0,Ci=1pc0 i
Q
i:Ri=0,Ci=0p
n
i.
E-step. The expectation of the complete data log-likelihood conditional on the observed
data and the parameter estimates pc0(k−1)
i and pn (k−1) i at the (k − 1)th step is Q(k) = E( n0 X i=1
[Ci(log pc0i + log ˜πc) + (1 − Ci){log pni + log(1 − ˜πc)}|Y1, . . . , Yn0, p
c0(k−1) i , pn (k−1) i ] = n0 X i=1
[Wi(k)(log pc0i + log ˜πc) + (1 − Wi(k)){log pni + log(1 − ˜πc)}]
where Wi(k) = pr(k−1)(C i = 1|Yi, Ri = 0, Ai = 0) = ˜πcpc0 (k−1) i /{˜πcpc0 (k−1) i + (1 − ˜πc)pn (k−1) i }.
M-step. We wish to maximize Q(k) over pc0
i , pni subject to (A1) with µn = ˆµn, πc = ˜πc.
We do this by conducting a grid search over µc0 =Pn0
i=1pc0i Yi. We now discuss maximizing
Q(k) given µc0= ˜µc0. We will denote the maximizing values of pc0
i , pni for µc0 = ˜µc0 by ˜pc0i , ˜pni.
Note that ˜µc0 is a possible value of µc0 if and only if
{pc0 i , i = 1, . . . , n0| X i pc0 i = 1, pc0i ≥ 0, X i pc0
i (Yi− µc0) = 0} is not empty. (A2)
For such a ˜µc0, maximizing Q(k)via Lagrange multipliers subject to (A1) and µc0 = ˜µc0gives
˜ pc0 i = Wi(k) (PiWi(k)){1 + ˜tc(Y i− ˜µc0)} , p˜n i = 1 − Wi(k) {Pi(1 − Wi(k))}{1 + ˜tn(Y i− ˆµn)}
where ˜tc and ˜tn can be determined in terms of ˜µc0 and ˆµn by
0 =X i ˜ pc0 i (Yi− ˜µc0) = X i Wi(k)(Yi− ˜µc0) (PiWi(k)){1 + ˜tc(Y i− ˜µc0)} (A3) 0 = X i ˜ pn i(Yi− ˆµn) = X i (1 − Wi(k))(Yi− ˆµn) {Pi(1 − Wi(k))}{1 + ˜tn(Y i− ˆµn)} (A4)
re-spectively, so that a safeguarded zero-finding algorithm, such as Brent’s method, can be used. Starting points for the zero finding algorithm can be found by noting that, since 0 ≤ ˜pc0 i , ˜pni ≤ 1, ˜tc∈ (1 − Wi(k) P iW (k) i ˜ µc0− Y (n0) , 1 − Wi(k) P iW (k) i ˜ µc0− Y (1) ) , ˜tn∈ ( 1 − (1−Wi(k)) P i(1−W (k) i ) ˆ µn− Y (n0) , 1 − (1−Wi(k)) P i(1−W (k) i ) ˆ µn− Y (1) )
where Y(n0) = max(Yi|Ri = 0) and Y(1) = min(Yi|Ri = 0). The kth-step parameter estimates
pc0(k)
i , pn
(k)
i , i = 1, . . . , n0, are the ˜pc0i , ˜pni that correspond to the ˜µc0 that maximizes Q(k) over
the grid of ˜µc0 considered. Note that we can avoid the need to consider the constraint (A2)
by replacing the logarithm function with the pseudo-logarithm function of Owen (2001, p. 62) in the definition of Q(k).
Appendix 2
Proofs
Outline proof of Lemma 1. The complete proof is provided in a technical report available
from the authors. Here we outline the steps in the proof.
Step 1. We show that maximizing Qn0
i=1qi subject to (6) and (8)-(10) with µn= ˆµn, πc=
˜
πc is a convex optimization problem so that there is a unique global maximum.
Step 2. Our problem involves maximization over a constrained parameter space.
Nettle-ton (1999) shows that, under regularity assumptions, the EM algorithm converges to either (a) a stationary point or (b) a boundary point of the constrained parameter space at which the likelihood function can be increased only by moving in a direction outside the param-eter space. For an unconstrained paramparam-eter space, under regularity assumptions, the EM
algorithm converges only to points of type (a) (Wu, 1983). We show that, even though our parameter space is constrained, under regularity assumptions, the EM algorithm converges only to points of type (a) for our problem.
Step 3. We combine the results in Steps 1 and 2 with results aboutEMfor unconstrained problems of Wu (1983) and Dempster et al. (1977) to prove the lemma.
Proof of Theorem 1. Let Z1, . . . , Zn0 denote the Y |R = 0 sample, and let ˆπcR=1 equal the
method of moments estimate of πc based on the R = 1 sample, ˆπR=1c = #{Ri = 1, Ai =
1}/(N −n0). Note that, if there exist pc0i , pni that satisfy (i) ˆπcR=1pc0i +(1−ˆπR=1c )pni = 1/n0, (ii)
Pn0
i=1pni(Zi− ˆµn) = 0, (iii)
Pn0
i=1pc0i =
Pn0
i=1pni = 1 and (iv) pci0, pni ≥ 0, then the approximate
maximum empirical likelihood estimator equals the standard instrumental variable estimator and the maximizing values of qi are qi = 1/n0, i = 1, . . . , n0. By considering the minimum
and maximum values of Pn0
i=1pniZi subject to (i), (iii) and (iv) above, we have that there
exist pc0
i , pni that satisfy (i)-(iv) if and only if ˆµn∈ [µl(N), µu(N)], where
µl(N) = bkXn0c i=1 Z(i) 1 kn0 + Z(bkn0c+1) kn0 − bkn0c kn0 , µu(N) = n0 X i=n0−bkn0c+1 Z(i) 1 kn0 + Z(n0−bkn0c) kn0 − bkn0c kn0 ,
kn0 = n0(1 − ˆπR=1c ) and bkc is the greatest integer less than or equal to k. Let ˜µl(N) and
˜
µu(N) be the trimmed sample means of Z1, . . . , Zn0 trimmed to the [0, 1 − πc] quantiles and
[πc, 1] quantiles respectively; that is,
˜ µl(N) = bn0(1−πXc)c i=1 Z(i) 1 1 − πc + Z(bn0(1−πc)c+1) n0(1 − πc) − bn0(1 − πc)c n0(1 − πc) .
Then, letting G denote the cumulative distribution function of the potential outcomes under the control, we have that, as N → ∞, in probability,
˜ µl(N) → 1 1 − πc Z G−1(1−πc) −∞ zdG(z) = µ∞ l , ˜ µu(N) → Z ∞ G−1(πc) zdG(z) = µ∞ u ,
by the properties of trimmed means (Shao, 2003, Ch. 5). Now we show that µl(N) → µ∞l
in probability and µu(N) → µ∞u in probability by showing that |µl(N) − ˜µl(N)| → 0 in
|µl(N) − ˜µl(N)| ≤ |s| max ¡ |Z(dn0(1−πc)+n0s+1e)|, |Z(bn0(1−πc)−n0s−1c)| ¢ (A5) where s = |(1 − ˆπR=1
c )−1− (1 − πc)−1| and dke is the least integer greater than or equal to k.
The first term on the right hand side of (A5) converges in probability to 0 as N → ∞ and the second term converges in probability to a number less than or equal to max(|G−1(1 −
πc + a)|, |G−1(1 − πc− a)|) for any number a > 0, for this, note that n0 = dN → ∞ as
N → ∞ since d > 0. This shows that the right-hand side, and hence the left hand side, of
(A5) converges in probability to 0 as N → ∞. Similarly,
|µu(N) − ˜µu(N)| ≤ |s| max
¡
|Z(dn0πc+n0s+1e)|, |Z(bn0πc−n0s−1c)|
¢
→ 0 in probability.
Thus, we conclude that µl(N) → µ∞l in probability and µu(N) → µ∞u in probability. By
assumption (13) that the distributions of compliers and never-takers overlap, we have that
µ∞
l < µn and µ∞u > µn. Combining the facts that µ∞l < µn < µ∞u , µl(N) → µ∞l in
probability and µu(N) → µ∞u in probability with the fact that ˆµn → µn in probability, by
the law of large numbers, because N − n0 = (1 − d)N → ∞ as N → ∞, we conclude that
pr{µl(N) < ˆµn < µu(N)} → 1 as N → ∞. Thus, pr(CACEˆ A =CACEˆ S) → 1 as N → ∞.
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Table 1: The relationship between observed groups and latent compliance classes Ri Ai Ci 1 1 1 (Complier) or 2 (Always-taker) 1 0 0 (Never-taker) or 3 (Defier) 0 0 0 (Never-taker) or 1 (Complier) 0 1 2 (Always-taker) or 3 (Defier)
Table 2: The relationship between observed groups and latent compliance classes in single consent design trials
Ri Ai Ci
1 1 1 (Complier)
1 0 0 (Never-taker)
0 0 1 (Complier) or 0 (Never-taker)
Table 3: The relationship between observed groups and latent compliance classes under Assumptions 1 − 6 Ri Ai Ci 1 1 1 (Complier) or 2 (Always-taker) 1 0 0 (Never-taker) 0 0 1 (Complier) or 0 (Never-taker) 0 1 2 (Always-taker)
Table 4: Estimates of theCACEwith true value 1 in single-consent treatment trials
Distn. N Bias Mean squared error
Std. IV AMELE Parametric Std. IV AMELE Parametric
N1 100 0.0178 −0.1141 −0.0240 0.3482 0.2003 0.1649 500 0.0202 −0.0016 −0.0054 0.0679 0.0515 0.0294 N2 100 0.0150 0.0105 −0.0053 0.1682 0.1604 0.1311 500 0.0019 0.0020 −0.0062 0.0214 0.0211 0.0186 G1 100 0.0429 −0.0981 0.2851 0.3697 0.1945 0.2424 500 −0.0060 −0.0212 0.3963 0.0637 0.0529 0.1907 G2 100 0.0088 −0.0048 0.3390 0.1957 0.1726 0.2311 500 0.0235 0.0232 0.3765 0.0454 0.0450 0.1561 LN1 100 0.0173 −0.1364 0.2299 0.2277 0.1008 0.1897 500 0.0177 −0.0266 0.3666 0.0411 0.0235 0.1568 LN2 100 −0.0007 −0.0137 0.2813 0.0670 0.0563 0.1593 500 0.0126 0.0129 0.2627 0.0120 0.0117 0.0814
Distn., distributions; Std. IV, standard instrumental variable estimate; AMELE, approximate maximum empirical likelihood estimate
Table 5: Results from the depression study
Hamilton score Composite anti-depression score Estimator estimate (95% CI) estimate (95% CI)
Std. IV −2.55(−4.13, −0.97) 1.86(0.76, 3.14)
AMELE −2.54(−4.12, −0.97) 1.60(0.73, 2.40) Parametric −2.82(−4.39, −1.16) 1.41(−0.66, 2.47)
CI, confidence interval; standard IV, standard instrumental variable estimate; AMELE, approximate maximum empirical likelihood estimate
−0.6 −0.4 −0.2 0.0 0.2 −10 −9 −8 −7 −6 −5
Complier average causal effect
Log−likelihood
Std IV
MLE
Figure 1: Profile log-likelihood for the maximum likelihood estimator and standard instru-mental variable estimator of the complier average causal effect for the sample described in
(a) Hamilton score Density 0 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 0.10 (b) Hamilton score Density 0 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 0.10 (c) Hamilton score Density 0 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 0.10 (d)
Composite anti−depression score
Density 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 (e)
Composite anti−depression score
Density 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 (f)
Composite anti−depression score
Density 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0
Figure 2: Depression study. Histograms of (a)-(c) the Hamilton score and (d)-(f) the com-posite anti-depression score for (a),(d) the R = 1, A = 1 group; (b), (e) the R = 1, A = 0 group; (c), (f) the R = 0 group
